Group #13136765 — subgroups

Subgroups — C₃×(C₂²⋊C₉)⋊C₁₆ #13136765

Order 1728.

138 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#4549	—	3	1	no	maximal
3	(C₂²⋊C₉)⋊C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×C₄×C₂²⋊C₉	C₄	1	1	yes	characteristic
5	C₃×C₉⋊C₁₆	—	4	1	no	maximal
6	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
7	C₂×C₃×C₃⋊C₁₆	—	3	1	no	—
8	C₂×C₃×C₃⋊C₁₆	—	3	1	no	—
9	C₈×C₂²⋊C₉	—	2	1	no	—
10	C₈×C₂²⋊C₉	C₂×C₃	1	1	yes	characteristic
11	C₂×C₃×C₂²⋊C₉	C₈	1	1	yes	characteristic
12	C₃×C₈×C₉	—	4	1	no	—
13	C₃×C₂²⋊C₁₆	—	9	1	no	—
14	#4549	—	3	1	no	—
15	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
16	C₂×C₃²×C₈	—	3	1	no	—
17	C₂×C₃²×C₈	—	3	1	no	—
18	C₄×C₂²⋊C₉	—	2	1	no	—
19	C₄×C₂²⋊C₉	C₃×C₄	1	1	yes	characteristic
20	C₃×C₃⋊C₁₆	—	6	1	no	—
21	C₃×C₃⋊C₁₆	—	6	1	no	—
22	C₉⋊C₁₆	—	4	1	no	—
23	C₃×C₂²⋊C₉	C₁₆	1	1	yes	characteristic, complemented
24	C₃×C₄×C₉	—	4	1	no	—
25	C₂²×C₃×C₈	—	2	1	no	—
26	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
28	C₂×C₃×C₁₆	—	9	1	no	—
29	C₂×C₃×C₁₆	—	9	1	no	—
30	C₂×C₃⋊C₁₆	—	3	1	no	—
31	C₂×C₃⋊C₁₆	—	3	1	no	—
32	C₂³×C₃²	C₃⋊C₈	1	1	yes	characteristic
33	C₂×C₃²×C₄	—	3	1	no	—
34	C₂×C₃²×C₄	—	3	1	no	—
35	C₂×C₂²⋊C₉	—	2	1	no	—
36	C₂×C₂²⋊C₉	C₃×C₈	1	1	yes	characteristic
37	C₃²×C₈	—	3	1	no	—
38	C₃²×C₈	S₄	1	1	yes	characteristic
39	C₈×C₉	—	4	1	no	—
40	C₈×C₉	—	8	1	no	—
41	C₂²⋊C₁₆	—	9	1	no	Sylow
42	C₂×C₃×C₉	—	4	1	no	—
43	C₂²×C₃×C₄	—	2	1	no	—
44	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
46	C₂×C₃×C₈	—	3	1	no	—
47	C₂×C₃×C₈	—	3	1	no	—
48	C₂×C₃×C₈	—	3	1	no	—
49	C₂×C₃×C₈	—	3	1	no	—
50	C₂×C₃×C₈	—	6	1	no	—
51	C₂×C₃×C₈	—	6	1	no	—
52	C₃×C₁₆	—	18	1	no	—
53	C₃×C₁₆	—	18	1	no	—
54	C₃⋊C₁₆	—	6	1	no	—
55	C₃⋊C₁₆	—	6	1	no	—
56	C₂²×C₃²	—	3	1	no	—
57	C₂²×C₃²	—	3	1	no	—
58	C₂²×C₃²	C₃⋊C₁₆	1	1	yes	characteristic
59	C₃²×C₄	—	3	1	no	—
60	C₃²×C₄	SL(2,ℤ/4)	1	1	yes	characteristic
61	C₂²⋊C₉	—	2	1	no	—
62	C₂²⋊C₉	C₃×C₁₆	1	1	yes	characteristic, complemented
63	C₄×C₉	—	4	1	no	—
64	C₄×C₉	—	8	1	no	—
65	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
66	C₂×C₁₆	—	9	1	no	—
67	C₂×C₁₆	—	9	1	no	—
68	C₃×C₉	—	4	1	no	Sylow
69	C₂³×C₃	—	2	1	no	—
70	C₂³×C₃	C₉⋊C₈	1	1	yes	characteristic
71	C₂³×C₃	C₃×C₃⋊C₈	1	1	yes	characteristic
72	C₂×C₃×C₄	—	3	1	no	—
73	C₂×C₃×C₄	—	3	1	no	—
74	C₂×C₃×C₄	—	3	1	no	—
75	C₂×C₃×C₄	—	3	1	no	—
76	C₂×C₃×C₄	—	6	1	no	—
77	C₂×C₃×C₄	—	6	1	no	—
78	C₃×C₈	—	2	1	no	—
79	C₃×C₈	—	3	1	no	—
80	C₃×C₈	—	3	1	no	—
81	C₃×C₈	—	6	1	no	—
82	C₃×C₈	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic
83	C₃×C₈	C₃×S₄	1	1	yes	characteristic
84	C₂×C₃²	—	3	1	no	—
85	C₂×C₃²	—	3	1	no	—
86	C₂×C₃²	A₄⋊C₈	1	1	yes	characteristic
87	C₂×C₉	—	4	1	no	—
88	C₂×C₉	—	8	1	no	—
89	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
90	C₂×C₈	—	3	1	no	—
91	C₂×C₈	—	3	1	no	—
92	C₁₆	—	18	1	no	—
93	C₁₆	—	18	1	no	—
94	C₂²×C₃	—	2	1	no	—
95	C₂²×C₃	—	3	1	no	—
96	C₂²×C₃	—	3	1	no	—
97	C₂²×C₃	—	3	1	no	—
98	C₂²×C₃	—	3	1	no	—
99	C₂²×C₃	—	6	1	no	—
100	C₂²×C₃	—	6	1	no	—
101	C₂²×C₃	C₉⋊C₁₆	1	1	yes	characteristic, complemented
102	C₂²×C₃	C₃×C₃⋊C₁₆	1	1	yes	characteristic
103	C₃×C₄	—	2	1	no	—
104	C₃×C₄	—	3	1	no	—
105	C₃×C₄	—	3	1	no	—
106	C₃×C₄	—	6	1	no	—
107	C₃×C₄	(C₂²⋊C₉)⋊C₄	1	1	yes	characteristic
108	C₃×C₄	C₃×SL(2,ℤ/4)	1	1	yes	characteristic
109	C₃²	A₄⋊C₁₆	1	1	yes	characteristic
110	C₉	—	4	1	no	—
111	C₉	—	8	1	no	—
112	C₂³	C₃×C₉⋊C₈	1	1	yes	characteristic
113	C₂×C₄	—	3	1	no	—
114	C₂×C₄	—	3	1	no	—
115	C₈	—	3	1	no	—
116	C₈	C₃×(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic
117	C₂×C₃	—	2	1	no	—
118	C₂×C₃	—	3	1	no	—
119	C₂×C₃	—	3	1	no	—
120	C₂×C₃	—	3	1	no	—
121	C₂×C₃	—	3	1	no	—
122	C₂×C₃	—	6	1	no	—
123	C₂×C₃	—	6	1	no	—
124	C₂×C₃	(C₂²⋊C₉)⋊C₈	1	1	yes	characteristic
125	C₂×C₃	C₃×A₄⋊C₈	1	1	yes	characteristic
126	C₂²	—	3	1	no	—
127	C₂²	—	3	1	no	—
128	C₂²	C₃×C₉⋊C₁₆	1	1	yes	characteristic, complemented
129	C₄	—	3	1	no	—
130	C₄	C₃×(C₂²⋊C₉)⋊C₄	1	1	yes	characteristic
131	C₃	—	2	1	no	—
132	C₃	(C₂²⋊C₉)⋊C₁₆	1	1	yes	characteristic, direct product factor
133	C₃	C₃×A₄⋊C₁₆	1	1	yes	characteristic
134	C₂	—	3	1	no	—
135	C₂	—	3	1	no	—
136	C₂	C₃×(C₂²⋊C₉)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₈×C₂²⋊C₉

C₂

yes

maximal, characteristic

C₃×#4549

—

maximal

(C₂²⋊C₉)⋊C₁₆

C₃

yes

maximal, characteristic, direct product factor

C₃×C₄×C₂²⋊C₉

C₄

yes

characteristic

C₃×C₉⋊C₁₆

—

maximal

C₂²×C₃²×C₈

D₃

yes

characteristic

C₂×C₃×C₃⋊C₁₆

—

C₂×C₃×C₃⋊C₁₆

—

C₈×C₂²⋊C₉

—

C₈×C₂²⋊C₉

C₂×C₃

yes

characteristic

C₂×C₃×C₂²⋊C₉

C₈

yes

characteristic

C₃×C₈×C₉

—

C₃×C₂²⋊C₁₆

—

#4549

—

C₂²×C₃²×C₄

Dic₃

yes

characteristic

C₂×C₃²×C₈

—

C₂×C₃²×C₈

—

C₄×C₂²⋊C₉

—

C₄×C₂²⋊C₉

C₃×C₄

yes

characteristic

C₃×C₃⋊C₁₆

—

C₃×C₃⋊C₁₆

—

C₉⋊C₁₆

—

C₃×C₂²⋊C₉

C₁₆

yes

characteristic, complemented

C₃×C₄×C₉

—

C₂²×C₃×C₈

—

C₂²×C₃×C₈

D₉

yes

characteristic

C₂²×C₃×C₈

C₃×D₃

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂³×C₃²

C₃⋊C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₂²⋊C₉

—

C₂×C₂²⋊C₉

C₃×C₈

yes

characteristic

C₃²×C₈

—

C₃²×C₈

S₄

yes

characteristic

C₈×C₉

—

C₈×C₉

—

C₂²⋊C₁₆

—

Sylow

C₂×C₃×C₉

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

Dic₉

yes

characteristic

C₂²×C₃×C₄

C₃×Dic₃

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃⋊C₁₆

yes

characteristic

C₃²×C₄

—

C₃²×C₄

SL(2,ℤ/4)

yes

characteristic

C₂²⋊C₉

—

C₂²⋊C₉

C₃×C₁₆

yes

characteristic, complemented

C₄×C₉

—

C₄×C₉

—

C₂²×C₈

C₃×D₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₃×C₉

—

Sylow

C₂³×C₃

—

C₂³×C₃

C₉⋊C₈

yes

characteristic

C₂³×C₃

C₃×C₃⋊C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

(C₂²⋊C₉)⋊C₂

yes

characteristic

C₃×C₈

C₃×S₄

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

A₄⋊C₈

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂²×C₄

C₃×Dic₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

—

101

C₂²×C₃

C₉⋊C₁₆

yes

characteristic, complemented

102

C₂²×C₃

C₃×C₃⋊C₁₆

yes

characteristic

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

(C₂²⋊C₉)⋊C₄

yes

characteristic

108

C₃×C₄

C₃×SL(2,ℤ/4)

yes

characteristic

109

C₃²

A₄⋊C₁₆

yes

characteristic

110

C₉

—

111

C₉

—

112

C₂³

C₃×C₉⋊C₈

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₈

—

116

C₈

C₃×(C₂²⋊C₉)⋊C₂

yes

characteristic

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂×C₃

—

121

C₂×C₃

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

(C₂²⋊C₉)⋊C₈

yes

characteristic

125

C₂×C₃

C₃×A₄⋊C₈

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

C₃×C₉⋊C₁₆

yes

characteristic, complemented

129

C₄

—

130

C₄

C₃×(C₂²⋊C₉)⋊C₄

yes

characteristic

131

C₃

—

132

C₃

(C₂²⋊C₉)⋊C₁₆

yes

characteristic, direct product factor

133

C₃

C₃×A₄⋊C₁₆

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

C₃×(C₂²⋊C₉)⋊C₈

yes

characteristic

	subgroup	G/H
center, hypercenter	C₃×C₈	(C₂²⋊C₉)⋊C₂
commutator, hypocenter	C₂²⋊C₉	C₃×C₁₆
Frattini	C₃×C₈	C₃×S₄
Fitting	C₂²×C₃²×C₈	D₃
socle, prime radical	C₂³×C₃²	C₃⋊C₈
cosocle	C₈×C₂²⋊C₉	C₂×C₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₃×C₈

subgroup

G/H

center, hypercenter

C₃×C₈

(C₂²⋊C₉)⋊C₂

commutator, hypocenter

C₂²⋊C₉

C₃×C₁₆

Frattini

C₃×C₈

C₃×S₄

Fitting

C₂²×C₃²×C₈

D₃

socle, prime radical

C₂³×C₃²

C₃⋊C₈

cosocle

C₈×C₂²⋊C₉

C₂×C₃

LCS

G ▷ C₂²⋊C₉

UCS

C₁ ◁ C₃×C₈

N	H ≅ G/N	type
3. (C₂²⋊C₉)⋊C₁₆	C₃	direct (N × H)
23. C₃×C₂²⋊C₉	C₁₆	semidirect
62. C₂²⋊C₉	C₃×C₁₆	semidirect
101. C₂²×C₃	C₉⋊C₁₆	semidirect
128. C₂²	C₃×C₉⋊C₁₆	semidirect
132. C₃	(C₂²⋊C₉)⋊C₁₆	direct (N × H)

H ≅ G/N

type

3. (C₂²⋊C₉)⋊C₁₆

C₃

direct (N × H)

23. C₃×C₂²⋊C₉

C₁₆

semidirect

62. C₂²⋊C₉

C₃×C₁₆

semidirect

101. C₂²×C₃

C₉⋊C₁₆

semidirect

128. C₂²

C₃×C₉⋊C₁₆

semidirect

132. C₃

(C₂²⋊C₉)⋊C₁₆

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₈×C₂²⋊C₉	2
C₃×#4549	3
(C₂²⋊C₉)⋊C₁₆	3
C₃×C₉⋊C₁₆	4

subgroup

index

C₃×C₈×C₂²⋊C₉

C₃×#4549

(C₂²⋊C₉)⋊C₁₆

C₃×C₉⋊C₁₆

Subgroups — C₃×(C₂²⋊C₉)⋊C₁₆ #13136765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×(C22⋊C9)⋊C16 #13136765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₂²⋊C₉)⋊C₁₆ #13136765